Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2015, Article ID 328273, 7 pages http://dx.doi.org/10.1155/2015/328273
Research Article Application of a Hybrid Method Combining Grey Model and Back Propagation Artificial Neural Networks to Forecast Hepatitis B in China Ruijing Gan, Xiaojun Chen, Yu Yan, and Daizheng Huang School of Preclinical Medicine, Guangxi Medical University, No. 22, Shuangyong Road, Nanning, Guangxi 530021, China Correspondence should be addressed to Daizheng Huang; [email protected] Received 20 September 2014; Revised 22 January 2015; Accepted 22 January 2015 Academic Editor: Chung-Min Liao Copyright © 2015 Ruijing Gan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Accurate incidence forecasting of infectious disease provides potentially valuable insights in its own right. It is critical for early prevention and may contribute to health services management and syndrome surveillance. This study investigates the use of a hybrid algorithm combining grey model (GM) and back propagation artificial neural networks (BP-ANN) to forecast hepatitis B in China based on the yearly numbers of hepatitis B and to evaluate the method’s feasibility. The results showed that the proposal method has advantages over GM (1, 1) and GM (2, 1) in all the evaluation indexes.
1. Introduction Hepatitis B is a vaccine preventable disease caused by the hepatitis B virus (HBV) that can induce potentially fatal liver damage. It has infected approximately 2 billion people worldwide, which represents one-third of the world population. Each year around the world, HBV infection is responsible for about one million deaths due to liver failure and cirrhosis and more than 75% of the hepatocellular carcinomas worldwide develop from HBV infection [1–3]. HBV is most prevalent in China, South East Asia, sub-Saharan Africa, and the Amazon basin of South America where health care resources are most limited [4]. In the Chinese population of 1.3 billion individuals, there are estimated to be 93 million HBV carriers. Each year, 300,000 deaths are attributed to chronic hepatitis B, including deaths associated with liver cirrhosis and hepatocellular carcinoma (HCC) [5]. Therefore, it is critical for early prevention of hepatitis B and an accurate forecasting which would enable public health officials to evaluate intervention strategies and make educated decisions. Mathematical and computational models have gained in importance in the public-health domain, especially in infectious disease epidemiology, by providing rationales and
quantitative analysis to support decision-making and policymaking processes in recent years. And many researchers advocate the use of these models as predictive tools [6–12]. The accurate forecasting of hepatitis B can be obtained by analyzing the sufficient historical data. However, in China and perhaps some other developing countries, the current public health surveillance system does not collect detailed essential epidemiological information as they are often difficult to obtain. The forecasted of hepatitis B will be inaccurate only by the limited data. Therefore, it is significant to make the limited data-processing. The grey systems theory chiefly including the theory of grey system analysis, modeling, prediction, decisionmaking, and control is established by Deng, which focuses on uncertainty problems with small samples, discrete data and incomplete information that are difficult for probability, and fuzzy mathematics to handle. Grey prediction is an important embranchment of grey systems theory, which makes scientific, quantitative forecasts about the future states of grey systems. The precise prediction of system can be performed by generating and extracting the useful information from the small samples and the partially known information [13–15].
2 Artificial neural networks (ANN) are complex and flexible nonlinear systems with properties not found in other modeling systems. It allows a method of forecasting with understanding of the relationship among variables and in particular nonlinear relationships. ANN function by initially learning a known set of data from a given problem with a known solution (training) and then the networks, inspired by the analytical processes of the human brain, are able to reconstruct the imprecise rules. Once a model is trained, the forecasted outputs can be generated from novel records [16– 19]. The aim of this study is to investigate the use of a hybrid method combining grey model (GM) and back propagation artificial neural networks (BP-ANN) to forecast hepatitis B in China based on the yearly numbers of hepatitis B from the years 2002 to 2012 and to evaluate the method’s performances of prediction.
Computational and Mathematical Methods in Medicine is defined as 𝑥0 (𝑡) + 𝑎𝑧1 (𝑡) = 𝑏, where −𝑎 and 𝑏 are referred to as the development coefficient and grey action quantity, respectively. Then −1
[𝑎𝑏]𝑇 = (𝐵1𝑇 𝐵1 ) 𝐵1𝑇 𝑌1 , where 𝑥0 (2) [ 0 ] [𝑥 (3)] [ ] ] 𝑌1 = [ [ .. ] , [ . ] [ ] 0 𝑥 (𝑛) [ ]
(2)
−𝑧1 (2) 1
[ 1 [−𝑧 (3) [ 𝐵1 = [ [ .. [ . [ 1 [−𝑧 (𝑛)
2. Materials and Methods 2.1. Data Sources. The incidence data of hepatitis B are collected from the Ministry of Health of the People’s Republic of China from the years 2002 to 2012, which are opening government statistics data [20]. 2.2. Methods. The proposed method is established based on the grey systems theory and BP-ANN theory. MATLAB software version 2011b is used for the statistical analysis. The incidence data are considered as the original time series 𝑋 = (𝑥0 , 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ), where 𝑛 is the length of the time series. Through grey generations or the effect of sequence operators to weaken the randomness, grey prediction models are designed to excavate the hidden laws; through the interchange between difference equations and differential equations, a practical jump of using discrete data sequences to establish continuous dynamic differential equations is materialized. Here, GM (1, 1) is the main and basic model of grey predictions, that is, a single variable first order grey model, which is able to acquire high prediction accuracy despite requiring small sample size (but the sample size must be at least 4). The GM (1, 1) model is suitable for sequences that show an obvious exponential pattern and can be used to describe monotonic changes. As for nonmonotonic wavelike development sequences, or saturated sigmoid sequences, one can consider establishing GM (2, 1) model. The establishment for a GM (1, 1) model is derived as follows. (1) Let nonnegative time sequence expressing 𝑋0 = 0 (𝑥 (1), 𝑥0 (2), . . . , 𝑥0 (𝑛)) be an original time sequence. Where 𝑛 is the sample size of the data. (2) First-order accumulative generation operation (1AGO) is used to convert 𝑋0 into 𝑋1 = (𝑥1 (1), 𝑥1 (2), . . . , 𝑥1 (𝑛)) = (∑1𝑖=1 𝑥0 (𝑖), ∑2𝑖=1 𝑥0 (𝑖), . . . , ∑𝑛𝑖=1 𝑥0 (𝑖)) (3) Let 𝑍1 = (𝑧1 (2), 𝑧1 (3), . . . , 𝑧1 (𝑛)) be the sequence generated from 𝑋1 by adjacent neighbor means. That is, 𝑧1 (𝑡) = 0.5(𝑥1 (𝑡) + 𝑥1 (𝑡 − 1)), 𝑡 = 2, 3, . . . , 𝑛. The least square estimate sequence of the grey difference equation of GM (1, 1)
(1)
𝑏.
] 1] ] ] .. ] . .] ]
1]
(4) The whitenization equation is given by 𝑑𝑥1 /𝑑𝑡+𝑎𝑥1 =
(5) The forecasting model can be obtained by solving the above equation, which is shown as follows: 𝑏 𝑏 𝑥̂1 (𝑡 + 1) = (𝑥0 (1) − ) 𝑒−𝑎𝑡 + . 𝑎 𝑎
(3)
(6) The predicted value of the primitive data at time point (𝑡 + 1) is extracted: 𝑥̂0 (𝑡 + 1) = 𝑥̂1 (𝑡 + 1) − 𝑥̂1 (𝑡) 𝑏 = (1 − 𝑒𝑎 ) (𝑥0 (1) − ) 𝑒−𝑎𝑡 . 𝑎
(4)
The procedure for a GM (2, 1) model is derived as follows. (1) For a given sequence of original data 𝑋0 = (𝑥0 (1), 𝑥0 (2), . . . , 𝑥0 (𝑛)), let its sequences of accumulation generation and inverse accumulation generation be 𝑋1 = (𝑥1 (1), 𝑥1 (2), . . . , 𝑥1 (𝑛)) and 𝛼1 𝑋0 = (𝛼1 𝑥0 (1), . . . , 𝛼1 𝑥0 (𝑛)), respectively, where 𝛼1 𝑥0 = 𝑥0 (𝑡) − 𝑥0 (𝑡 − 1), 𝑡 = 2, 3, . . . , 𝑛 and the sequence of adjacent neighbor mean generation of 𝑋1 is 𝑍1 = (𝑧1 (2), 𝑧1 (3), . . . , 𝑧1 (𝑛)). (2) The GM (2, 1) model is 𝛼1 𝑥0 (𝑡) + 𝑎1 𝑥0 (𝑡) + 𝑎2 𝑧1 (𝑡) = 𝑢 and the whitenization equation is given by 𝑑2 𝑥1 /𝑑𝑡2 + 𝛼1 (𝑑𝑥1 /𝑑𝑡) + 𝛼2 𝑥1 = 𝑢. The least squares estimate of the parametric sequence is 𝑇
−1
𝑎̂ = [𝑎1 , 𝑎2 , 𝑢] = (𝐵2𝑇 𝐵2 ) 𝐵2𝑇 𝑌2 ,
(5)
Computational and Mathematical Methods in Medicine
3
where
The original time series of hepatitis B
0
1
−𝑥 (2) −𝑧 (2) 1 ] [ 0 [−𝑥 (3) −𝑧1 (3) 1 ] ] [ 𝐵2 = [ ]. ] [ ⋅⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ] [ 0 1 −𝑧 1 −𝑥 (𝑛) (𝑛) ] [
GM (1,1)
The prediction
(3) Solve the whitenization equation. If 𝑋1∗ is a special 1 solution of the whitenization equation and 𝑋 the general solution of the corresponding homogeneous equation 1 𝑑2 𝑥1 /𝑑𝑡2 +𝛼1 (𝑑𝑥1 /𝑑𝑡)+𝛼2 𝑥1 = 0, then 𝑋1∗ +𝑋 is the general solution of the GM (2, 1) whitenization equation. There are three cases for the general solution of the homogeneous equation: (i) when the characteristic equation 𝑟2 + 𝛼1 𝑟 + 1 𝛼2 = 0 has two distinct real roots 𝑋 = 𝑐1 𝑒𝑟1 𝑡 + 𝑐2 𝑒𝑟2 𝑡 ; (ii) 1 when the characteristic equation has a repeated root 𝑋 = 𝑟𝑡 𝑒 (𝑐1 + 𝑐2 𝑡); (iii) when the characteristic equation has two complex conjugate roots 𝑟1 = 𝛼 + 𝑖𝛽 and 𝑟2 = 𝛼 − 𝑖𝛽, 1 𝑋 = 𝑒𝛼𝑡 (𝑐1 cos 𝛽𝑡 + 𝑐2 sin 𝛽𝑡). A special solution of the whitenization equation may take of the three possibilities: (i) when 0 is not a root of the characteristic equation, 𝑋1∗ = 𝐶; (ii) when 0 is one of the two distinct roots of the characteristic equation, 𝑋1∗ = 𝐶𝑥; and (iii) when 0 is the only root of the characteristic equation, 𝑋1∗ = 𝐶𝑥2 . The steps of the forecasting method can be described as follows. Step 1 (train the BP-ANN). In order to obtain the input of the BP-ANN, the GM (1, 1) and GM (2, 1) model are used to predict for the original time series of hepatitis B, respectively. The two groups of prediction are taken as the input of the BP-ANN. At the same time, the original time series are taken as the output. Thus the structure of a three-layer BP-ANN is constructed and the trained BP-ANN model will be obtained by training. Step 2 (forecast by the trained BP-ANN). The GM (1, 1) and GM (2, 1) model are used to forecast for the original time series of hepatitis B, respectively, at first. Then the two groups of forecasted data are taken as the input of the trained BPANN. Finally, the forecasted of hepatitis B will be obtained by running the trained BP-ANN. The method flow chart is shown in Figure 1.
GM (2,1)
The prediction
GM (1,1)
GM (2,1)
The forecasted
The training BP-ANN
(6)
The forecasted
The trained BP-ANN
The input layer
The input layer
The hidden layer
The hidden layer
The output layer (the original data)
The output layer (the forecasted)
Figure 1: Flow chart of the hybrid method.
The incidence number of hepatitis B
𝛼1 𝑥0 (2) 𝑥0 (2) − 𝑥0 (1) [ 1 0 ] [ 0 ] [𝛼 𝑥 (3)] [ 𝑥 (3) − 𝑥0 (2) ] [ ] [ ] 𝑌2 = [ ]=[ ], [ ⋅⋅⋅ ] [ ] ⋅ ⋅ ⋅ [ ] [ ] 1 0 0 0 − 𝑥 − 1) 𝛼 𝑥 𝑥 (𝑛) (𝑛) (𝑛 [ ] [ ]
×105 12
11 10 9 8 7 6 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time
Figure 2: The incidence number of hepatitis B in China from 2002 to 2012.
̂ mean square error 𝑅𝑦𝑦 ̂ is the covariance between 𝑦 and 𝑦; 2
(MSE), MSE = (1/𝑛)∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) ; root mean square error 2
(RMSE); RMSE = √(1/𝑛)∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) ; mean average error (MAE), MAE = (1/𝑛) ∑𝑛𝑖=1 |𝑦̂𝑖 − 𝑦𝑖 |; mean average percentage error (MAPE), MAPE = (1/𝑛) ∑𝑛𝑖=1 |𝑦̂𝑖 − 𝑦𝑖 | ∗ 100, and sum 2 of squared error (SSE), SSE = ∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) .
4. Result The incidence data of hepatitis B are collected year by year from 2002 to 2012 in China and taken as the original time series, which is shown in Figure 2.
3. Evaluation Criteria The metrics used are relative error (RE), RE𝑖 = |𝑦̂𝑖 − 𝑦𝑖 |/𝑦𝑖 , 𝑖 = 1, 2, . . . , 𝑛, where 𝑦̂𝑖 is the forecasted data and 𝑦𝑖 is the actual ̂ where data; correlation coefficient (𝑅), 𝑅 = 𝑅𝑦𝑦 ̂ /std(𝑦)std(𝑦),
4.1. The GM (1, 1) and GM (2, 1) Models Calculation. The GM (1, 1) and GM (2, 1) models are calculated and shown as follows.
4
Computational and Mathematical Methods in Medicine
Iutput layer
The incidence number of hepatitis B
Hidden layer 1 1 Output layer . ..
2
1
5
×106 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 2012 2013 2014 2015 2016 2017 2018 2019 2010 2021
Year
Figure 3: The topology structure of the proposal method.
Figure 4: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the proposal method.
4.1.1. The Parameters GM (1, 1) Model. Consider the following: −𝑎 = 0.027523, 𝑏 = 893075.402372. Therefore, the GM (1, 1) model of this time series can be forecasted for long term forecasting for the reasons that GM (1, 1) can be used for long term forecasting when −𝑎 ≤ 0.3 and for short term forecasting when 0.3 < −𝑎 ≤ −0.5. −𝑎 reflects the development states of the accumulation generated sequence 𝑥̂(1) and the sequence of raw data 𝑥̂(0) .
×106 1.3
GM (2, 1) Model. Consider the following: 𝑎1 = 0.4083, 𝑎2 = 0.0158, 𝑢 = 583425.10336, 𝑟1 = −0.04329, 𝑟2 = −0.36502, 𝑐1 = −38764735.45099, and 𝑐2 = 2527585.93078.
0.8
1.2 1.1 1 0.9
0.7
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12 ×105
GM (1,1) GM (2,1) The proposal method
4.1.2. The Forecasting Models GM (1, 1) Model. Consider the following:
Figure 5: The scatter diagram of the relationship between the observed data and the prediction.
0.027523𝑡
𝑥 (𝑡 + 1) = −33116202.802344𝑒
+ 32447876.802344.
(7)
GM (2, 1) Model. Consider the following: 𝑥1 (𝑡 + 1) = −38764735.45099𝑒−0.04329𝑡 + 2527585.93078𝑒−0.36502𝑡
(8)
+ 36918146.77020. 4.2. The Forecasted. In the three-layer BP-ANN, the hidden node 𝑛2 and the input node 𝑛1 are related by 𝑛2 = 2𝑛1 + 1. The two groups of prediction created by the two GM models are taken as the input of the BP-ANN and the observed data is taken as the output. Therefore, a three-layer proposed model with 2 input nodes, 5 hidden nodes, and 1 output node is obtained. The topology structure is shown in Figure 3. The weights and thresholds of the proposal model will be obtained by training. Let the training time be 1000, the learning rate be 0.9, the momentum factor be 0.95, and the error be 0.001; Levenberg Marquardt is used as training algorithm.
The prediction of the original time series by the GM (1, 1) and GM (2, 1) model, respectively, are taken as the input of the trained BP-ANN. Then the forecasted of hepatitis B will be obtained by running the trained BP-ANN. The forecasted is shown in Figure 4.
5. Discussion In order to compare the prediction created by the two GM models and the proposed method, a prediction is performed under the same conditions. The results are listed in Table 1 and the scatter diagram is shown in Figure 5. It can be seen from Figure 5 that the prediction generated by the two GM models has greater dispersion than that by the proposed method. The RE of prediction is shown in Figure 6. From the figure, we know that the prediction obtained by the proposed method has higher accuracy and smaller RE than that by the GM approaches. Figure 6 indicates that the smaller the relative error is, the closer prediction is to the observed data.
Computational and Mathematical Methods in Medicine
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Table 1: The prediction created by the GM (1, 1), GM (2, 1), and the proposal model. Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
The observed data 719011 916396 982297 1109130 1169946 1169569 1179607 1060582 1093335 1087086
GM (1, 1) 924130 949918 976426 1003674 1031682 1060471 1090065 1120484 1151751 1183892
GM (2, 1) 882183 1036306 1133758 1183850 1201811 1198178 1180237 1153018 1119982 1083509
The proposal method 736600 884300 1027700 1096300 1118200 1124300 1125900 1126400 1126500 1126500
Table 2: The evaluation indexes comparison. Index The proposal method GM (1, 1) model GM (2, 1) model
𝑅 0.9495 0.6365 0.9392
MSE 2.3649 × 107 2.2867 × 108 1.6798 × 108
MAE 3.9704 × 104 1.0492 × 106 1.1173 × 106
Table 3: The forecasted generated by the three methods. GM (1, 1)
GM (2, 1)
The proposal method
2013 2014 2015 2016 2017 2018 2019 2020 2021
1216929.0 1250888.2 1285795.1 1321676.0 1358558.3 1396469.7 1435439.1 1475496.0 1516670.7
1045223.6 1006228.6 967267.6 928834.0 891248.9 854715.0 819353.3 785229.0 752369.4
1077864.1 1074038.2 1012371.7 976301.8 946959.7 939194.1 937881.5 937607.7 937531.5
The comparison of 𝑅, MSE, MAE, RMSE, MAPE, and SSE are listed in Table 2. It can be seen that the proposed method has advantages over GMs in all the evaluation indexes. The forecasted generated by the GM (1, 1), GM (2, 1), and the proposal model are listed in Table 3 and shown in Figure 7. The weights and thresholds of BP-ANN will generate randomly at first when the model is training. This will make the predicted and forecasted uncertainty. To describe this clearer, the proposal model is ran 100 times and the mean value will be taken as predicted or forecasted value. The 95% confidence interval and predicted or forecasted value are shown in Figures 8 and 9, respectively. Although the prediction result created by the proposal method in the paper has more accurate than that by the two gray models, the proposal model has its limitations. Firstly, since the proposal model is built on the basis of gray model, the sample size, namely, the number of historical data must be not less than 4. Secondly, the prediction result
MAPE 3.9704 × 106 1.0492 × 108 1.1173 × 108
SSE 1.8162 × 1010 1.1078 × 1013 1.2570 × 1013
1.4 1.2 The relative error
Year
RMSE 4.863 × 103 1.5122 × 104 1.5122 × 104
1 0.8 0.6 0.4 0.2 0
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time GM (1,1) GM (2,1) The proposal method
Figure 6: Comparison of the RE of the prediction by the proposal method and the GMs.
will be inaccuracy if the weights and thresholds in BPANN ran into local optimum in the process of training. Intelligent algorithms can be used to optimize the weights and thresholds of BP-ANN [21].
6. Conclusion The hepatitis B epidemiological information is often difficult to obtain. Forecasting of hepatitis B will be inaccurate by the limited data. The grey systems theory focuses on uncertainty problems with small samples and incomplete information. At the same time, the BP-ANN is a method of forecasting with understanding of the relationship among variables and nonlinear relationships. The research proposes a new forecasting method, which combines the GM and BP-ANN, to forecast hepatitis B in China. The useful information can generate and
Computational and Mathematical Methods in Medicine
The incidence number of hepatitis B
6 ×106 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Year
The proposal method GM (1,1) GM (2,1)
The incidence number of hepatitis B
Figure 7: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the three methods. ×105 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time
The predicted value The confidence interval
The incidence number of hepatitis B
Figure 8: The predicted incidence of hepatitis B in China from 2003 to 2012 by the proposal methods.
×106 1.1
1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 2013
2014
2015
2016
2017
2018
2019
2020
2021
Time The forecasted value The confidence interval
Figure 9: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the proposal methods.
extract from the small samples and the BP neural networks can train data more sufficiently. The prediction results show that this method can obtain better forecasting.
Conflict of Interests The authors have declared that no competing interests exist.
Acknowledgments The authors thank Zola Banh for carefully correcting grammar errors. This work was supported by Youth Science Foundation of Guangxi Medical University (GXMUYSF201208), the open project of Guangxi Medical Science Experimental Center (KFJJ2011-31), the training programs of innovation and entrepreneurship for undergraduates in Guangxi province (2012xjcxcy006), and the training programs of innovation and entrepreneurship for undergraduates in China (201210598002). The funders had no role in study design, data collection, and analysis, decision to publish, or preparation of the paper.
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Research Article Application of a Hybrid Method Combining Grey Model and Back Propagation Artificial Neural Networks to Forecast Hepatitis B in China Ruijing Gan, Xiaojun Chen, Yu Yan, and Daizheng Huang School of Preclinical Medicine, Guangxi Medical University, No. 22, Shuangyong Road, Nanning, Guangxi 530021, China Correspondence should be addressed to Daizheng Huang; [email protected] Received 20 September 2014; Revised 22 January 2015; Accepted 22 January 2015 Academic Editor: Chung-Min Liao Copyright © 2015 Ruijing Gan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Accurate incidence forecasting of infectious disease provides potentially valuable insights in its own right. It is critical for early prevention and may contribute to health services management and syndrome surveillance. This study investigates the use of a hybrid algorithm combining grey model (GM) and back propagation artificial neural networks (BP-ANN) to forecast hepatitis B in China based on the yearly numbers of hepatitis B and to evaluate the method’s feasibility. The results showed that the proposal method has advantages over GM (1, 1) and GM (2, 1) in all the evaluation indexes.
1. Introduction Hepatitis B is a vaccine preventable disease caused by the hepatitis B virus (HBV) that can induce potentially fatal liver damage. It has infected approximately 2 billion people worldwide, which represents one-third of the world population. Each year around the world, HBV infection is responsible for about one million deaths due to liver failure and cirrhosis and more than 75% of the hepatocellular carcinomas worldwide develop from HBV infection [1–3]. HBV is most prevalent in China, South East Asia, sub-Saharan Africa, and the Amazon basin of South America where health care resources are most limited [4]. In the Chinese population of 1.3 billion individuals, there are estimated to be 93 million HBV carriers. Each year, 300,000 deaths are attributed to chronic hepatitis B, including deaths associated with liver cirrhosis and hepatocellular carcinoma (HCC) [5]. Therefore, it is critical for early prevention of hepatitis B and an accurate forecasting which would enable public health officials to evaluate intervention strategies and make educated decisions. Mathematical and computational models have gained in importance in the public-health domain, especially in infectious disease epidemiology, by providing rationales and
quantitative analysis to support decision-making and policymaking processes in recent years. And many researchers advocate the use of these models as predictive tools [6–12]. The accurate forecasting of hepatitis B can be obtained by analyzing the sufficient historical data. However, in China and perhaps some other developing countries, the current public health surveillance system does not collect detailed essential epidemiological information as they are often difficult to obtain. The forecasted of hepatitis B will be inaccurate only by the limited data. Therefore, it is significant to make the limited data-processing. The grey systems theory chiefly including the theory of grey system analysis, modeling, prediction, decisionmaking, and control is established by Deng, which focuses on uncertainty problems with small samples, discrete data and incomplete information that are difficult for probability, and fuzzy mathematics to handle. Grey prediction is an important embranchment of grey systems theory, which makes scientific, quantitative forecasts about the future states of grey systems. The precise prediction of system can be performed by generating and extracting the useful information from the small samples and the partially known information [13–15].
2 Artificial neural networks (ANN) are complex and flexible nonlinear systems with properties not found in other modeling systems. It allows a method of forecasting with understanding of the relationship among variables and in particular nonlinear relationships. ANN function by initially learning a known set of data from a given problem with a known solution (training) and then the networks, inspired by the analytical processes of the human brain, are able to reconstruct the imprecise rules. Once a model is trained, the forecasted outputs can be generated from novel records [16– 19]. The aim of this study is to investigate the use of a hybrid method combining grey model (GM) and back propagation artificial neural networks (BP-ANN) to forecast hepatitis B in China based on the yearly numbers of hepatitis B from the years 2002 to 2012 and to evaluate the method’s performances of prediction.
Computational and Mathematical Methods in Medicine is defined as 𝑥0 (𝑡) + 𝑎𝑧1 (𝑡) = 𝑏, where −𝑎 and 𝑏 are referred to as the development coefficient and grey action quantity, respectively. Then −1
[𝑎𝑏]𝑇 = (𝐵1𝑇 𝐵1 ) 𝐵1𝑇 𝑌1 , where 𝑥0 (2) [ 0 ] [𝑥 (3)] [ ] ] 𝑌1 = [ [ .. ] , [ . ] [ ] 0 𝑥 (𝑛) [ ]
(2)
−𝑧1 (2) 1
[ 1 [−𝑧 (3) [ 𝐵1 = [ [ .. [ . [ 1 [−𝑧 (𝑛)
2. Materials and Methods 2.1. Data Sources. The incidence data of hepatitis B are collected from the Ministry of Health of the People’s Republic of China from the years 2002 to 2012, which are opening government statistics data [20]. 2.2. Methods. The proposed method is established based on the grey systems theory and BP-ANN theory. MATLAB software version 2011b is used for the statistical analysis. The incidence data are considered as the original time series 𝑋 = (𝑥0 , 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ), where 𝑛 is the length of the time series. Through grey generations or the effect of sequence operators to weaken the randomness, grey prediction models are designed to excavate the hidden laws; through the interchange between difference equations and differential equations, a practical jump of using discrete data sequences to establish continuous dynamic differential equations is materialized. Here, GM (1, 1) is the main and basic model of grey predictions, that is, a single variable first order grey model, which is able to acquire high prediction accuracy despite requiring small sample size (but the sample size must be at least 4). The GM (1, 1) model is suitable for sequences that show an obvious exponential pattern and can be used to describe monotonic changes. As for nonmonotonic wavelike development sequences, or saturated sigmoid sequences, one can consider establishing GM (2, 1) model. The establishment for a GM (1, 1) model is derived as follows. (1) Let nonnegative time sequence expressing 𝑋0 = 0 (𝑥 (1), 𝑥0 (2), . . . , 𝑥0 (𝑛)) be an original time sequence. Where 𝑛 is the sample size of the data. (2) First-order accumulative generation operation (1AGO) is used to convert 𝑋0 into 𝑋1 = (𝑥1 (1), 𝑥1 (2), . . . , 𝑥1 (𝑛)) = (∑1𝑖=1 𝑥0 (𝑖), ∑2𝑖=1 𝑥0 (𝑖), . . . , ∑𝑛𝑖=1 𝑥0 (𝑖)) (3) Let 𝑍1 = (𝑧1 (2), 𝑧1 (3), . . . , 𝑧1 (𝑛)) be the sequence generated from 𝑋1 by adjacent neighbor means. That is, 𝑧1 (𝑡) = 0.5(𝑥1 (𝑡) + 𝑥1 (𝑡 − 1)), 𝑡 = 2, 3, . . . , 𝑛. The least square estimate sequence of the grey difference equation of GM (1, 1)
(1)
𝑏.
] 1] ] ] .. ] . .] ]
1]
(4) The whitenization equation is given by 𝑑𝑥1 /𝑑𝑡+𝑎𝑥1 =
(5) The forecasting model can be obtained by solving the above equation, which is shown as follows: 𝑏 𝑏 𝑥̂1 (𝑡 + 1) = (𝑥0 (1) − ) 𝑒−𝑎𝑡 + . 𝑎 𝑎
(3)
(6) The predicted value of the primitive data at time point (𝑡 + 1) is extracted: 𝑥̂0 (𝑡 + 1) = 𝑥̂1 (𝑡 + 1) − 𝑥̂1 (𝑡) 𝑏 = (1 − 𝑒𝑎 ) (𝑥0 (1) − ) 𝑒−𝑎𝑡 . 𝑎
(4)
The procedure for a GM (2, 1) model is derived as follows. (1) For a given sequence of original data 𝑋0 = (𝑥0 (1), 𝑥0 (2), . . . , 𝑥0 (𝑛)), let its sequences of accumulation generation and inverse accumulation generation be 𝑋1 = (𝑥1 (1), 𝑥1 (2), . . . , 𝑥1 (𝑛)) and 𝛼1 𝑋0 = (𝛼1 𝑥0 (1), . . . , 𝛼1 𝑥0 (𝑛)), respectively, where 𝛼1 𝑥0 = 𝑥0 (𝑡) − 𝑥0 (𝑡 − 1), 𝑡 = 2, 3, . . . , 𝑛 and the sequence of adjacent neighbor mean generation of 𝑋1 is 𝑍1 = (𝑧1 (2), 𝑧1 (3), . . . , 𝑧1 (𝑛)). (2) The GM (2, 1) model is 𝛼1 𝑥0 (𝑡) + 𝑎1 𝑥0 (𝑡) + 𝑎2 𝑧1 (𝑡) = 𝑢 and the whitenization equation is given by 𝑑2 𝑥1 /𝑑𝑡2 + 𝛼1 (𝑑𝑥1 /𝑑𝑡) + 𝛼2 𝑥1 = 𝑢. The least squares estimate of the parametric sequence is 𝑇
−1
𝑎̂ = [𝑎1 , 𝑎2 , 𝑢] = (𝐵2𝑇 𝐵2 ) 𝐵2𝑇 𝑌2 ,
(5)
Computational and Mathematical Methods in Medicine
3
where
The original time series of hepatitis B
0
1
−𝑥 (2) −𝑧 (2) 1 ] [ 0 [−𝑥 (3) −𝑧1 (3) 1 ] ] [ 𝐵2 = [ ]. ] [ ⋅⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ] [ 0 1 −𝑧 1 −𝑥 (𝑛) (𝑛) ] [
GM (1,1)
The prediction
(3) Solve the whitenization equation. If 𝑋1∗ is a special 1 solution of the whitenization equation and 𝑋 the general solution of the corresponding homogeneous equation 1 𝑑2 𝑥1 /𝑑𝑡2 +𝛼1 (𝑑𝑥1 /𝑑𝑡)+𝛼2 𝑥1 = 0, then 𝑋1∗ +𝑋 is the general solution of the GM (2, 1) whitenization equation. There are three cases for the general solution of the homogeneous equation: (i) when the characteristic equation 𝑟2 + 𝛼1 𝑟 + 1 𝛼2 = 0 has two distinct real roots 𝑋 = 𝑐1 𝑒𝑟1 𝑡 + 𝑐2 𝑒𝑟2 𝑡 ; (ii) 1 when the characteristic equation has a repeated root 𝑋 = 𝑟𝑡 𝑒 (𝑐1 + 𝑐2 𝑡); (iii) when the characteristic equation has two complex conjugate roots 𝑟1 = 𝛼 + 𝑖𝛽 and 𝑟2 = 𝛼 − 𝑖𝛽, 1 𝑋 = 𝑒𝛼𝑡 (𝑐1 cos 𝛽𝑡 + 𝑐2 sin 𝛽𝑡). A special solution of the whitenization equation may take of the three possibilities: (i) when 0 is not a root of the characteristic equation, 𝑋1∗ = 𝐶; (ii) when 0 is one of the two distinct roots of the characteristic equation, 𝑋1∗ = 𝐶𝑥; and (iii) when 0 is the only root of the characteristic equation, 𝑋1∗ = 𝐶𝑥2 . The steps of the forecasting method can be described as follows. Step 1 (train the BP-ANN). In order to obtain the input of the BP-ANN, the GM (1, 1) and GM (2, 1) model are used to predict for the original time series of hepatitis B, respectively. The two groups of prediction are taken as the input of the BP-ANN. At the same time, the original time series are taken as the output. Thus the structure of a three-layer BP-ANN is constructed and the trained BP-ANN model will be obtained by training. Step 2 (forecast by the trained BP-ANN). The GM (1, 1) and GM (2, 1) model are used to forecast for the original time series of hepatitis B, respectively, at first. Then the two groups of forecasted data are taken as the input of the trained BPANN. Finally, the forecasted of hepatitis B will be obtained by running the trained BP-ANN. The method flow chart is shown in Figure 1.
GM (2,1)
The prediction
GM (1,1)
GM (2,1)
The forecasted
The training BP-ANN
(6)
The forecasted
The trained BP-ANN
The input layer
The input layer
The hidden layer
The hidden layer
The output layer (the original data)
The output layer (the forecasted)
Figure 1: Flow chart of the hybrid method.
The incidence number of hepatitis B
𝛼1 𝑥0 (2) 𝑥0 (2) − 𝑥0 (1) [ 1 0 ] [ 0 ] [𝛼 𝑥 (3)] [ 𝑥 (3) − 𝑥0 (2) ] [ ] [ ] 𝑌2 = [ ]=[ ], [ ⋅⋅⋅ ] [ ] ⋅ ⋅ ⋅ [ ] [ ] 1 0 0 0 − 𝑥 − 1) 𝛼 𝑥 𝑥 (𝑛) (𝑛) (𝑛 [ ] [ ]
×105 12
11 10 9 8 7 6 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time
Figure 2: The incidence number of hepatitis B in China from 2002 to 2012.
̂ mean square error 𝑅𝑦𝑦 ̂ is the covariance between 𝑦 and 𝑦; 2
(MSE), MSE = (1/𝑛)∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) ; root mean square error 2
(RMSE); RMSE = √(1/𝑛)∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) ; mean average error (MAE), MAE = (1/𝑛) ∑𝑛𝑖=1 |𝑦̂𝑖 − 𝑦𝑖 |; mean average percentage error (MAPE), MAPE = (1/𝑛) ∑𝑛𝑖=1 |𝑦̂𝑖 − 𝑦𝑖 | ∗ 100, and sum 2 of squared error (SSE), SSE = ∑𝑛𝑖=1 (𝑦̂𝑖 − 𝑦𝑖 ) .
4. Result The incidence data of hepatitis B are collected year by year from 2002 to 2012 in China and taken as the original time series, which is shown in Figure 2.
3. Evaluation Criteria The metrics used are relative error (RE), RE𝑖 = |𝑦̂𝑖 − 𝑦𝑖 |/𝑦𝑖 , 𝑖 = 1, 2, . . . , 𝑛, where 𝑦̂𝑖 is the forecasted data and 𝑦𝑖 is the actual ̂ where data; correlation coefficient (𝑅), 𝑅 = 𝑅𝑦𝑦 ̂ /std(𝑦)std(𝑦),
4.1. The GM (1, 1) and GM (2, 1) Models Calculation. The GM (1, 1) and GM (2, 1) models are calculated and shown as follows.
4
Computational and Mathematical Methods in Medicine
Iutput layer
The incidence number of hepatitis B
Hidden layer 1 1 Output layer . ..
2
1
5
×106 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 2012 2013 2014 2015 2016 2017 2018 2019 2010 2021
Year
Figure 3: The topology structure of the proposal method.
Figure 4: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the proposal method.
4.1.1. The Parameters GM (1, 1) Model. Consider the following: −𝑎 = 0.027523, 𝑏 = 893075.402372. Therefore, the GM (1, 1) model of this time series can be forecasted for long term forecasting for the reasons that GM (1, 1) can be used for long term forecasting when −𝑎 ≤ 0.3 and for short term forecasting when 0.3 < −𝑎 ≤ −0.5. −𝑎 reflects the development states of the accumulation generated sequence 𝑥̂(1) and the sequence of raw data 𝑥̂(0) .
×106 1.3
GM (2, 1) Model. Consider the following: 𝑎1 = 0.4083, 𝑎2 = 0.0158, 𝑢 = 583425.10336, 𝑟1 = −0.04329, 𝑟2 = −0.36502, 𝑐1 = −38764735.45099, and 𝑐2 = 2527585.93078.
0.8
1.2 1.1 1 0.9
0.7
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12 ×105
GM (1,1) GM (2,1) The proposal method
4.1.2. The Forecasting Models GM (1, 1) Model. Consider the following:
Figure 5: The scatter diagram of the relationship between the observed data and the prediction.
0.027523𝑡
𝑥 (𝑡 + 1) = −33116202.802344𝑒
+ 32447876.802344.
(7)
GM (2, 1) Model. Consider the following: 𝑥1 (𝑡 + 1) = −38764735.45099𝑒−0.04329𝑡 + 2527585.93078𝑒−0.36502𝑡
(8)
+ 36918146.77020. 4.2. The Forecasted. In the three-layer BP-ANN, the hidden node 𝑛2 and the input node 𝑛1 are related by 𝑛2 = 2𝑛1 + 1. The two groups of prediction created by the two GM models are taken as the input of the BP-ANN and the observed data is taken as the output. Therefore, a three-layer proposed model with 2 input nodes, 5 hidden nodes, and 1 output node is obtained. The topology structure is shown in Figure 3. The weights and thresholds of the proposal model will be obtained by training. Let the training time be 1000, the learning rate be 0.9, the momentum factor be 0.95, and the error be 0.001; Levenberg Marquardt is used as training algorithm.
The prediction of the original time series by the GM (1, 1) and GM (2, 1) model, respectively, are taken as the input of the trained BP-ANN. Then the forecasted of hepatitis B will be obtained by running the trained BP-ANN. The forecasted is shown in Figure 4.
5. Discussion In order to compare the prediction created by the two GM models and the proposed method, a prediction is performed under the same conditions. The results are listed in Table 1 and the scatter diagram is shown in Figure 5. It can be seen from Figure 5 that the prediction generated by the two GM models has greater dispersion than that by the proposed method. The RE of prediction is shown in Figure 6. From the figure, we know that the prediction obtained by the proposed method has higher accuracy and smaller RE than that by the GM approaches. Figure 6 indicates that the smaller the relative error is, the closer prediction is to the observed data.
Computational and Mathematical Methods in Medicine
5
Table 1: The prediction created by the GM (1, 1), GM (2, 1), and the proposal model. Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
The observed data 719011 916396 982297 1109130 1169946 1169569 1179607 1060582 1093335 1087086
GM (1, 1) 924130 949918 976426 1003674 1031682 1060471 1090065 1120484 1151751 1183892
GM (2, 1) 882183 1036306 1133758 1183850 1201811 1198178 1180237 1153018 1119982 1083509
The proposal method 736600 884300 1027700 1096300 1118200 1124300 1125900 1126400 1126500 1126500
Table 2: The evaluation indexes comparison. Index The proposal method GM (1, 1) model GM (2, 1) model
𝑅 0.9495 0.6365 0.9392
MSE 2.3649 × 107 2.2867 × 108 1.6798 × 108
MAE 3.9704 × 104 1.0492 × 106 1.1173 × 106
Table 3: The forecasted generated by the three methods. GM (1, 1)
GM (2, 1)
The proposal method
2013 2014 2015 2016 2017 2018 2019 2020 2021
1216929.0 1250888.2 1285795.1 1321676.0 1358558.3 1396469.7 1435439.1 1475496.0 1516670.7
1045223.6 1006228.6 967267.6 928834.0 891248.9 854715.0 819353.3 785229.0 752369.4
1077864.1 1074038.2 1012371.7 976301.8 946959.7 939194.1 937881.5 937607.7 937531.5
The comparison of 𝑅, MSE, MAE, RMSE, MAPE, and SSE are listed in Table 2. It can be seen that the proposed method has advantages over GMs in all the evaluation indexes. The forecasted generated by the GM (1, 1), GM (2, 1), and the proposal model are listed in Table 3 and shown in Figure 7. The weights and thresholds of BP-ANN will generate randomly at first when the model is training. This will make the predicted and forecasted uncertainty. To describe this clearer, the proposal model is ran 100 times and the mean value will be taken as predicted or forecasted value. The 95% confidence interval and predicted or forecasted value are shown in Figures 8 and 9, respectively. Although the prediction result created by the proposal method in the paper has more accurate than that by the two gray models, the proposal model has its limitations. Firstly, since the proposal model is built on the basis of gray model, the sample size, namely, the number of historical data must be not less than 4. Secondly, the prediction result
MAPE 3.9704 × 106 1.0492 × 108 1.1173 × 108
SSE 1.8162 × 1010 1.1078 × 1013 1.2570 × 1013
1.4 1.2 The relative error
Year
RMSE 4.863 × 103 1.5122 × 104 1.5122 × 104
1 0.8 0.6 0.4 0.2 0
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time GM (1,1) GM (2,1) The proposal method
Figure 6: Comparison of the RE of the prediction by the proposal method and the GMs.
will be inaccuracy if the weights and thresholds in BPANN ran into local optimum in the process of training. Intelligent algorithms can be used to optimize the weights and thresholds of BP-ANN [21].
6. Conclusion The hepatitis B epidemiological information is often difficult to obtain. Forecasting of hepatitis B will be inaccurate by the limited data. The grey systems theory focuses on uncertainty problems with small samples and incomplete information. At the same time, the BP-ANN is a method of forecasting with understanding of the relationship among variables and nonlinear relationships. The research proposes a new forecasting method, which combines the GM and BP-ANN, to forecast hepatitis B in China. The useful information can generate and
Computational and Mathematical Methods in Medicine
The incidence number of hepatitis B
6 ×106 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Year
The proposal method GM (1,1) GM (2,1)
The incidence number of hepatitis B
Figure 7: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the three methods. ×105 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Time
The predicted value The confidence interval
The incidence number of hepatitis B
Figure 8: The predicted incidence of hepatitis B in China from 2003 to 2012 by the proposal methods.
×106 1.1
1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 2013
2014
2015
2016
2017
2018
2019
2020
2021
Time The forecasted value The confidence interval
Figure 9: The forecasted incidence of hepatitis B in China from 2013 to 2021 by the proposal methods.
extract from the small samples and the BP neural networks can train data more sufficiently. The prediction results show that this method can obtain better forecasting.
Conflict of Interests The authors have declared that no competing interests exist.
Acknowledgments The authors thank Zola Banh for carefully correcting grammar errors. This work was supported by Youth Science Foundation of Guangxi Medical University (GXMUYSF201208), the open project of Guangxi Medical Science Experimental Center (KFJJ2011-31), the training programs of innovation and entrepreneurship for undergraduates in Guangxi province (2012xjcxcy006), and the training programs of innovation and entrepreneurship for undergraduates in China (201210598002). The funders had no role in study design, data collection, and analysis, decision to publish, or preparation of the paper.
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